{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R664",
  "content_hash": null,
  "slug": "rm28-claim-relative-cubic-radius-88",
  "type": "claim",
  "title": "The relative cubic covering radius equals 88",
  "summary": "The relative covering radius of RM(2,8) inside RM(3,8) equals 88. The full covering radius maximizes over every 8-variable Boolean function and may be larger.",
  "relevance": "For Covering radius of the second-order Reed-Muller code RM(2,8), record rm28-claim-relative-cubic-radius-88 (“The relative cubic covering radius equals 88”) records a bound, answer, status fact, or structural consequence. The record states: The relative covering radius of RM(2,8) inside RM(3,8) equals 88.",
  "relevance_source": "recorded",
  "body": "Khoruzhii, Gelß, and Pokutta define\n\\[\n\\rho_{2,3}(m)=\\max_{F\\in RM(3,m)}d_2(F)\n\\]\nand record \\(\\rho_{2,3}(8)=88\\), based on Hou's complete classification of cubic forms through eight variables. Since \\(RM(3,8)\\) is a subclass of all 8-variable Boolean functions, this value supplies the lower bound \\(\\rho(2,8)\\geq88\\). Global equality remains the canonical target. The distinction between the relative maximum and the full maximum prevents the cubic classification from being presented as a complete solution.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "family",
    "statement": "all 8-variable Boolean functions of degree at most three",
    "family": "RM(3,8), viewed modulo RM(2,8)"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2607.02365v1",
      "locator": "Section 2, equations defining d_r and the relative radius; discussion of rho_{2,3}(8)=88"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2607.02365v1",
    "locator": "Section 2, equations defining d_r and the relative radius; discussion of rho_{2,3}(8)=88"
  },
  "relations": [
    {
      "slug": "R662",
      "title": "The full covering radius satisfies 88 <= rho(2,8) <= 96",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R660",
      "title": "A 2026-07-28 source audit confirms the current 88 to 96 interval",
      "object_type": "attempt",
      "relation": "reports",
      "direction": "incoming"
    },
    {
      "slug": "R659",
      "title": "Reproduce the B(3,4,7) high-nonlinearity orbit classification",
      "object_type": "attempt",
      "relation": "uses",
      "direction": "incoming"
    },
    {
      "slug": "reed-muller-rm2-8-covering-radius",
      "title": "reed muller rm2 8 covering radius",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}